Problem 1 - The Calculus with Analytic Geometry

Question

In Exercises 1 through 20 , find the derivative of the given function. $$ F(x)=\left(x^{2}+4 x-5\right)^{3} $$

Step-by-Step Solution

Answer

\( F'(x) = 3(x^2 + 4x - 5)^2 (2x + 4) \)

Step 1: Understand the Problem

We need to find the derivative of the function \( F(x) = (x^2 + 4x - 5)^3 \). This requires using the chain rule.

Step 2: Apply the Chain Rule

The chain rule states that if you have a composite function \( h(g(x)) \), then the derivative is \( h'(g(x)) \times g'(x) \). Identify the outer function and the inner function. Here, the outer function \( h(u) = u^3 \) and the inner function is \( g(x) = x^2 + 4x - 5 \).

Step 3: Differentiate the Outer Function

Differentiate the outer function \( h(u) = u^3 \) with respect to \( u \). This gives us \( h'(u) = 3u^2 \).

Step 4: Differentiate the Inner Function

Differentiate the inner function \( g(x) = x^2 + 4x - 5 \) with respect to \( x \). This yields \( g'(x) = 2x + 4 \).

Step 5: Combine Results Using the Chain Rule

According to the chain rule, multiply the derivative of the outer function by the derivative of the inner function: \[ F'(x) = 3(g(x))^2 \times g'(x) \]. Substitute \( g(x) \) and \( g'(x) \) back into the equation: \[ F'(x) = 3(x^2 + 4x - 5)^2 \times (2x + 4) \].

Step 6: Simplify the Result

Distribute and simplify if necessary. The simplified derivative is: \[ F'(x) = 3(x^2 + 4x - 5)^2 (2x + 4) \].

Key Concepts

Derivative

In calculus, derivatives represent how a function changes as its input changes. It is essentially the slope of the function at a given point and is foundational for understanding rates of change. If we denote a function as \(f(x)\), its derivative is denoted as \(f'(x)\) or \(\frac{df(x)}{dx}\). To find the derivative, we apply specific rules and techniques, such as the power rule, product rule, and the chain rule. Understanding derivatives helps in tackling more complex problems involving motion, growth, and various physical phenomena.

Chain Rule

The chain rule is a vital technique in calculus for finding the derivative of a composite function. A composite function is one that entails one function inside another, such as \( h(g(x)) \). The chain rule states that the derivative of \( h(g(x)) \) is \( h'(g(x)) \times g'(x) \). This means you first differentiate the outer function, leaving the inner function unchanged, and then multiply by the derivative of the inner function. This rule is crucial for dealing with functions nested within others, as seen in our original exercise.

Composite Function

A composite function occurs when one function is applied to the result of another function. For instance, if \( g(x) \) is an inner function and \( h(u) \) is an outer function, the composite function is written as \( h(g(x)) \). In practical terms, composite functions allow more complex modeling of real-world scenarios, where one process affects another. In our specific exercise, \( F(x) = (x^2 + 4x - 5)^3 \), \( x^2 + 4x - 5 \) is the inner function, and raising it to the power of 3 is the outer function. Understanding and handling composite functions are essential skills in calculus.